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\begin{document}

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\centerline{\textbf{\large Exam \#3 $\cdot$ Thursday Nov. 10, 2005}}
\vspace*{.2in}
MATH 110 $\cdot$ Section 10 $\cdot$ Fall 2005  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

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\textbf{Problem 1.}  Let $$f(x) = \frac{2(x-1)(x+2)}{(x+1)(x-2)}.$$

(a) (\textbf{5 points}) The $x$-intercept(s) of $f(x)$ are:

\begin{tabular}{lll}
(A) 1 and -2 \qquad\qquad &
(B) -1 and 2 \qquad\qquad &
(C) 1, -1, 2, and -2 \qquad\qquad \\
(D) $f(x)$ has no $x$-intercepts &
(E) None of these \\
\end{tabular}

\vspace*{1.0in}

(b) (\textbf{5 points}) The vertical asymptote(s) of $f(x)$ are:

\begin{tabular}{lll}
(A) 1 and -2 \qquad\qquad &
(B) -1 and 2 \qquad\qquad &
(C) 1, -1, 2, and -2 \qquad\qquad \\
(D) $f(x)$ has no vertical asymptotes &
(E) None of these \\
\end{tabular}

\vspace*{1.0in}

(c) (\textbf{5 points}) Find an equation for the extreme behavior (i.e.
horizontal or slant asymptote) of $f(x)$.
\vspace*{1.0in}

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\textbf{Problem 2.} Let $$ g(x) = \frac{3(x^2+2x+1)} {x^2-1}.$$

(a) (\textbf{5 points}) True or false:  $g(x)$ has a vertical asymptote
at $x=1$.

\vspace*{1.0in}

(b) (\textbf{5 points}) True or false:  $g(x)$ has a vertical asymptote
at $x=-1$.

\vspace*{1.0in}

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\newpage
\textbf{Problem 3.} (\textbf{6 points})  The following is a graph of the
equation $y = kb^x$.  Find the values of $k$ and $b$.

% xxx label points (0, 3), (2, 3/4)

\vspace*{2.0in}

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\textbf{Problem 4.}  You invested \$800 in a savings account at your hometown
bank in May of 2005.  The interest rate is 3\%, compounded monthly.  Due in
part to your outstanding success in math 110, you are on track to graduate in
May of 2009.

(a) (\textbf{6 points}) What will be your balance when you graduate?

\vspace*{1.5in}

(b) (\textbf{6 points}) After how many years (to the nearest 0.1 year) will
your balance be \$1000?  (Hint:  There is an algebraic way to solve this
problem and a graphical way.  Use either one.)

\vspace*{1.5in}

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\textbf{Problem 5.} (\textbf{6 points})  After graduation, you plan to purchase
a new Ford Fiesta to drive to your new job in Connecticut.  Projected cost in
November of 2009 is \$12,000.  It is now November of 2005.  Assuming a local
bank has annual interest rate of 4\% compounded continuously, how much should
you invest now in order to reach your desired goal?

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\newpage
\textbf{Problem 6.}
(\textbf{6 points})
A baked potato is heated to $50^\circ$C and placed on the counter.
From that time, its temperature $A(t)$ in degrees Celsius, with time $t$
measured in hours, is described by the equation $$ A(t) = 22 +
(50-22)e^{-0.41t}.$$
What is the limiting temperature of the potato after an indefinitely large
amount of time goes by?

\begin{tabular}{lll}
(A) $50^\circ$C \qquad\qquad &
(B) $22^\circ$C \qquad\qquad &
(C) $28^\circ$C \qquad\qquad \\
(D) Cannot be determined &
(E) None of these \\
from the above information
\end{tabular}

\vspace*{1.0in}

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\textbf{Problem 7.} (\textbf{6 points})  Which of the following functions are
one-to-one?

\vspace*{1.5in}

%Incl. a rational function and a piecewise one with gappy domain.

\begin{tabular}{lll}
(A) 1 and 2 only \qquad\qquad &
(B) 1 and 3 only \qquad\qquad &
(C) 2 only \qquad\qquad \\
(D) 2 and 3 only &
(E) All of them \\
\end{tabular}

\vspace*{1.5in}

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\textbf{Problem 8.} (\textbf{6 points})  Let $$h(x) = \frac{2x+3}{x-4}.$$
Algebraically find the inverse function $h^{-1}(x)$.

\vspace*{2.0in}

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\newpage
\textbf{Problem 9.} Let $g(x) = \log(x+3)-2$.

(a) (\textbf{5 points}) Find the domain of $g(x)$.

\vspace*{1.5in}

(b) (\textbf{5 points}) Find the $x$-intercept(s) and $y$-intercept(s) of
$g(x)$, if any, to the nearest 0.01.

\vspace*{1.5in}

(c) (\textbf{5 points}) Find the horizontal and vertical asymptote(s) of
$g(x)$, if any.

\vspace*{1.5in}

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\textbf{Problem 10.} (\textbf{6 points}) Expand the following expression as
much as possible:  $$\log\left(\frac{ax^2+bx+c}{d^3}\right).$$

%% ----------------------------------------------------------------
\newpage
\textbf{Problem 11.} (\textbf{6 points})  Solve the following equation for $x$:
$$ \log(x+3) - 2 = 4.$$

\vspace*{2.0in}

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\textbf{Problem 12.} (\textbf{6 points}) The radioactvity of a mysterious
substance found in a beaker is given by the equation
$$ R(t) = 2.8 e^{-0.4t}$$
where $t$ is measured in hours and $R(t)$ measured in units of rems.  Find (to
within the nearest 0.01) the time $t$ at which $R(t)$ is one fourth of its
original amount $R(0)$.

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\end{document}
